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Find the indefinite integral. (Note: Solve by the simplest method—not all require integration by parts. Use C for the constant of integration.)?

#int t ln (t+3)dt#

Answer 1

I would say by parts is easier as you have simple steps to follow, but this is another way to get to the answer which is #ln(3)[(t^2 ln(t))/2 - t^2/4].#

#inttln(t+3)dt = ln(3)inttln(t)dt# - by rules of logarithms

Now, note how the derivative of #tln(t)# is #ln(t) + 1# But we need #tln(t)#. If you look at the derivative of #t^2ln(t)# it is #2tln(t) + t#

So #int2tln(t)dt + t = t^2ln(t)# #=> int2tln(t)dt = t^2ln(t) - inttdt# #=> inttln(t)dt = t^2ln(t)/2 - 1/2 inttdt# #=> inttln(t)dt =t^2ln(t)/2 - t^2/4 + C# #=> ln(3)inttln(t)dt =ln(3)[t^2ln(t)/2 - t^2/4] + C#

Hope this helps.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.